My research interests include computability theory, inductive inference and grammatical inference. Within computability theory, I have a particular interest in degree theory and applications of computability theory in other fields. My research thus far in inductive inference has been applications of computability theory to answer questions relevant to objects in inductive inference. Grammatical inference is concerned with the identification of generative grammars for languages.

We examine sets of codes such that certain properties are invariant under the choice of oracle from a range of possible oracles and establish a connection between such codes and Medvedev reductions. In examing the complexity of such sets of \emph{universal codes}, we prove completeness results at various levels of the arithmetic hierarchy as well as two general theorems for obtaining Π11-completeness for sets of universal codes. Among other corollaries, we show that the set of codes for Medvedev reductions of bi-immune sets to DNC functions is Π11-complete.

This paper classifies the complexity of various teaching models by their position in the arithmetical hierarchy. In particular, we determine the arithmetical complexity of the index sets of the following classes: (1) the class of uniformly r.e. families with finite teaching dimension, and (2) the class of uniformly r.e. families with finite positive recursive teaching dimension witnessed by a uniformly r.e. teaching sequence. We also derive the arithmetical complexity of several other decision problems in teaching.

Effective Bi-Immunity and Randomness

(To appear in the proceedings of Computability and Complexity Symposium 2017)

We study the relationship between randomness and effective bi-immunity. Greenberg and Miller have shown that for any oracle X, there are arbitrarily slow-growing DNR functions relative to X that compute no ML random set. We show that the same holds when ML randomness is replaced with effective bi-immunity. It follows that there are sequences of effective Hausdorff dimension 1 that compute no effectively bi-immune set.
We also establish an important difference between the two properties. The class Low(MLR, EBI) of oracles relative to which every ML random is effectively bi-immune contains the jump-traceable sets, and is therefore of cardinality continuum.

We compare the efficiency of learning using a membership oracle and learning using a teacher. We find that there is a large class of families for which the teacher provides a super-exponential efficiency gain over learning with an oracle. Furthermore, using both a teacher and an oracle provides a super-exponential gain over either learning with only an oracle and learning with only a teacher.

We prove the existence of a canonical form for semi-deterministic transducers with incomparable sets of output strings. Based on this, we develop an algorithm which learns semi-deterministic transducers given access to translation queries. We also prove that there is no learning algorithm for semi-deterministic transducers that uses only domain knowledge.

We prove the existence of a canonical form for semi-deterministic transducers with incomparable sets of output strings. Based on this, we develop an algorithm which learns semi-deterministic transducers given access to translation queries. We also prove that there is no learning algorithm for semi-deterministic transducers that uses only domain knowledge.

Given any oracle, A, we construct a basic sequence Q, computable in the jump of A, such that no A-computable real is Q-distribution-normal. A corollary to this is that there is a Δ0n+1 basic sequence with respect to which no Δ0n real is distribution-normal. As a special case, there is a limit computable sequence relative to which no computable real is distribution-normal.

In Diagonally Non-Computable Functions and Bi-Immunity, Carl Jockusch and Andrew Lewis proved that every DNC function computes a bi-immune set. They asked whether every DNC function computes an effectively bi-immune set. We construct a DNC function that computes no effectively bi-immune set, thereby answering their question in the negative.

In 1986, Osherson, Stob and Weinstein asked whether two variants of anomalous vacillatory learning, TxtFex** and TxtFext**, could be distinguished. These learning criteria place bounds neither on the number of hypotheses between which the learner is allowed to vacillate nor on the number of errors permitted, merely that both are finite. The criteria differ in that the more restrictive one, TxtFext**-learning, requires that all hypotheses output infinitely often must describe the same finite variant of the correct set, while TxtFex** permits the learner to vacillate between finitely many different finite variants of the correct set. In this paper we show that TxtFex** ≠ TxtFext**, thereby answering the question posed by Osherson, et al. We prove this in a strong way by exhibiting a family in TxtFex*2\TxtFext**.

We consider the arithmetic complexity of index sets of uniformly computably enumerable families learnable under different learning criteria. We determine the exact complexity for the standard notions of finite learning, learning in the limit, behaviourally correct learning and anomalous learning in the limit. In proving the Σ05-completeness result for behaviourally correct learning we prove a result of independent interest; if a u.c.e. family is not learnable, for any computable learner there is a Δ02 enumeration witnessing failure.